Monte Carlo and Quasi-Monte Carlo Methods for Statistics

本项目扩展并改进了蒙特卡罗采样技术。主要思想是将起源于准蒙特卡罗采样的思想纳入蒙特卡罗采样。特别是这个项目嵌入准蒙特卡罗采样到马尔可夫链蒙特卡罗模拟，直到最近还不知道是可能的组合。这种组合可以非常有效，在某些问题中使方差减少100倍以上。项目人员正在确定准蒙特卡罗在什么时候给马尔可夫链蒙特卡罗带来了很大的改进，以及寻找新的方法来结合这些方法。该项目改进蒙特卡罗采样的另一个领域是无界函数的集成。有一些版本的随机拟蒙特卡罗抽样比原来的拟蒙特卡罗抽样有更好的收敛率，至少对于表现良好的积分。不幸的是，无界积分的问题并没有得到很大的改善。本课题将随机拟蒙特卡罗抽样的优点扩展到无界被积，通过应用变量变换公式来约束被积，同时努力防止被积变得太尖。该项目还研究了加速马尔可夫链蒙特卡罗混合的调温方法，以及蒙特卡罗和准蒙特卡罗思想在生物信息学问题中的应用。蒙特卡罗采样几乎用于科学和工程的每个分支。最简单的方法是使用随机数生成器模拟系统，并记录发生的情况。在实践中，蒙特卡罗方法被用来用蛮力计算来解决一些难以用数学方法解决的问题。现实世界的复杂性可以很容易地引入到模拟中，通常会使问题难以进行精确的数学处理。准蒙特卡罗方法可以用来驱动模拟字节，用非常仔细地选择的比随机数更平衡的数字代替随机数序列。其结果通常是给定精度水平的极大加速，或者给定计算时间的精度的极大提高。这个项目将准蒙特卡罗方法推进到迄今为止被认为无法从中受益的模拟中。这些模拟技术被称为马尔科夫链蒙特卡罗，被用于许多领域，包括材料科学、教育测试数据集分析、生物医学研究、机器人技术、计算机图形学和市场营销。该项目还在寻找其他方法来改进蒙特卡罗方法，这些方法具有同样广泛的潜在效益。

This project extends and improves Monte Carlo sampling techniques. The main idea is to incorporate ideas that originated in quasi-Monte Carlo sampling into Monte Carlo sampling. In particular this project embeds quasi-Monte Carlo sampling into Markov chain Monte Carlo simulations, a combination that until recently was not known to be possible. The combination can be very effective, bringing variance reductions of over 100 fold in some problems. The project personnel are identifying when quasi-Monte Carlo brings a large improvement in Markov chain Monte Carlo, as well as finding new ways to combine the methods. Another area in which this project is improving Monte Carlo sampling is in integration of unbounded functions. There are versions of randomized quasi-Monte Carlo sampling that attain a better convergence rate than the original quasi-Monte Carlo sampling, at least for well behaved integrands. Unfortunately, problems with unbounded integrands don't see much improvement. This project extends the benefit of randomized quasi-Monte Carlo sampling to unbounded integrands by applying a change of variable formula to bound the integrand while endeavoring to prevent the integrand from becoming too spiky. This project is also investigating tempering methods for speeding up the mixing of Markov chain Monte Carlo as well as applications of Monte Carlo and quasi-Monte Carlo ideas to problems in bioinformatics.Monte Carlo sampling is used in just about every branch of science and engineering. At its simplest it involves simulating a system using random number generators, and recording what happens. In practice Monte Carlo methods are used to solve by brute force computation some problems that are too hard to do mathematically. Real world complications that can easily be introduced into a simulation often make a problem too hard for exact mathematical treatment. Quasi-Monte Carlo methods can be used to drive a simulation byt replacing the random number sequence by very carefully chosen numbers that are much more balanced than random numbers are. The result is often a tremendous speedup for a given level of accuracy, or a tremendous increase in accuracy for a given computation time. This project pushes quasi-Monte Carlo methods into simulations that had hitherto been thought incapable of benefiting from them. Those simulation techniques, known as Markov chain Monte Carlo, are used in many areas including materials science, analysis of educational testing data sets, biomedical research, robotics, computer graphics, and marketing. This project is also looking at other ways to improve Monte Carlo methods with similarly broad potential for benefit.